When they lined up 7 in a row, there were 6 soldiers left over. Some logical operators are not truth-functional.

However, whenever a shorter wff is used in constructing a more complicated wff, the parentheses on the shorter wff are necessary.

He's been called the best scientist of the Middle Ages; his Book of Optics has been called the most important physics text prior to Newton; his writings in physics anticipate the Principle of Least Action, Newton's First Law of Motion, and the notion that white light is composed of the color spectrum.

Classical or "bivalent" truth-functional propositional logic is that branch of truth-functional propositional logic that assumes that there are are only two possible truth-values a statement whether simple or complex can have: It is due to these paradoxes that the use of infinitesimals, which provides the basis for mathematical analysis, has been regarded as a non-rigorous heuristic and is finally viewed as sound only after the work of the great 19th-century rigorists, Dedekind and Weierstrass.

If instead my understanding of your explanation and the WP article's description of "ergative" is incorrect, and "ergative" just means any old verb that can be used transitively with object A and intransitively with subject A, then this label does not convey anything meaningful in English grammar contexts that is not better and more commonly expressed by simply stating that the verb in question is both transitive and intransitive c.

He requested that a representation of such a sphere and cylinder be inscribed on his tomb. We only care about how terms are actually used.

Consider a number n and let q be the lowest common multiple of the numbers p-1 for all the prime factors p of n. Archytas is sometimes called the "Father of Mathematical Mechanics.

The notion of a "truth table" is often utilized in the discussion of truth-functional connectives discussed below. History The serious study of logic as an independent discipline began with the work of Aristotle BCE.

The numerical subscripts are used just in case we need to deal with more than 26 simple statements: In our metalanguage, we shall also be using certain variables that are used to stand for arbitrary expressions built from the basic symbols of PL.

There are ingenious solutions available with other tools. I left the pronunciation addition alone. There is a slight distinction in meaning, though, which relates to the implication of an agent.

I suppose that means we should strip xml: Such an action that occurs by itself without a clear agent is called mediopassive or middle voiceand has a very strong connection with the reflexive. Many, many English verbs could meet this quite loose definition of "ergative" as given above.

Others claim these were first seen years earlier in Chang Tshang's Chinese text and were implicit in what survives of earlier Hindu works, but Brahmagupta's text discussed them lucidly. Apollonius soon surpassed it, but by using Archimedes' method.

Earlier Hindus, including Brahmagupta, contributed to this method. Similarly for freezesinkgrowetc. For a sentence like "the boat sinksthere is no actor, no agent that is making this happen. But it's not a context template, obviously. It is said that the discovery of irrational numbers upset the Pythagoreans so much they tossed Hippasus into the ocean!

In a famous leap of over-confidence he claimed he could control the Nile River; when the Caliph ordered him to do so, he then had to feign madness! The Greek emphasis on pure mathematics and proofs was key to the future of mathematics, but they were missing an even more important catalyst: Aristarchus would be almost unknown except that Archimedes mentions, and assumes, Aristarchus' heliocentrism in The Sand Reckoner.

Perhaps they used it to make a right-angled triangle so they could make true right-angles when constructing buildings - we do not know for certain. Swadesh lists but I found it, I didn't write it.

Moreover, hunters and herders employed the concepts of one, two, and many, as well as the idea of none or zero, when considering herds of animals.

Leonardo provided Europe with the decimal system, algebra and the 'lattice' method of multiplication, all far superior to the methods then in use. Aryabhata made several important discoveries in astronomy, e.

There is some evidence that Aristotleor at least his successor at the LyceumTheophrastus d.The Chinese Remainder Theorem Evan Chen∗ February 3, The Chinese Remainder Theorem is a \theorem" only in that it is useful and requires proof.

The name "Chinese Remainder Theorem" comes from 19th century Europe, see What is the history of the name “Chinese remainder theorem”? For a more mathematically focused history see Kangsheng's Historical development of the Chinese remainder theorem.

Chinese remainder theorem The Chinese remainder theorem describes an important class of linear Diophantine systems of equations: let n 1,n k be k pairwise coprime integers greater than one, a 1,a k be k arbitrary integers, and N be the product n 1 ··· n k.

Mathematica» The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha» Explore anything with the first computational knowledge engine. The Chinese Remainder Theorem is a number theoretic result. It is one of the only theorems named for an oriental person or place, due to the closed development of mathematics in the western world.

Chinese Remainder Theorem SHEN KANGSHENG Communicated by C. TRUESDELL 1. Source of the Problem This statement is called the SuN Z~ Theorem, or the Chinese Remainder Theo- rem. Indeed, in imitation of the theorem, YANa History df Hindu Mathematics,Lahole, Vol.

2, pp.

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